78 Algebra. 



Represent the equation bv 



/- = A'. (I) 



Multiply both members by 1\ obtaining 



LT=RT. (2) 



Now this may be written as 



(L-R)T^o. (3) 



If T is a known quantits' this can only be satisfied by the sup- 

 position that L = R, that is, the equation is equivalent to ( i). 

 But if 7' is a function of the unknown (for example, 2x or x+s, 

 or j;^+8) then (3) may be satisfied by any value of the unknown 

 that \vill make 7 = o (such as x=o, or a==5, or .t== — 2, respect- 

 ively, in the three examples given), whence (3) would not be 

 equivalent to (i) but to the two equations. 



( r=o. I 



6. Corollary. If am' equation involves fractions ivith 07ily 

 kjioivn quayitities in the denominators, it is legitimate to clear of 



fractions. The multiplier in this case is a known quantity. 



7, TheorKM. //* an equation involves irreducible fractions zvith 

 unknown quantities in the denominators, a?id the denominators are 

 all prime to each other, it is legitimate to ititegralize by multiplying 

 through by the least common multiple of the de?iominators. 



To illustrate the reasoning take the equation 



I 2 3 

 where the fractions are supposed to be in their lowest teniis and 

 Jfj, X ^, X ^ represent diffei^ent functions of the unknown and 

 where A, B and C are either known quantities or functions of 

 the unknown. Multiplying by the least common multiple of the 

 denominators we obtain 



AXX^-^BXX^-^CXX^=^o, (2) 



Now% since X , X^ and X ^ are prime to each other no common 

 factor has been introduced by multiplying by X X^X^, and con- 

 sequently no additional solutions can appear. 



